Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-6x-3y &= 6 \\ 2x-3y &= -6\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}6x+3y &= -6\\ 2x-3y &= -6\end{align*}$ Add the top and bottom equations. $8x = -12$ Divide both sides by $8$ and reduce as necessary. $x = -\dfrac{3}{2}$ Substitute $-\dfrac{3}{2}$ for $x$ in the top equation. $-6( -\dfrac{3}{2})-3y = 6$ $9-3y = 6$ $-3y = -3$ $y = 1$ The solution is $\enspace x = -\dfrac{3}{2}, \enspace y = 1$.